Finite Group Databasebeta

← C2×#32123 #7507750

Subgroups — C2×#32123 #7507750

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2489C211yesmaximal, characteristic
2C2×#2492C211yesmaximal, characteristic
3#32123C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic
5C2×#2056C211yesmaximal, characteristic, complemented
6C2×#2056C211yesmaximal, characteristic, complemented
7C2×#2026C211yesmaximal, characteristic, complemented
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2001C211yesmaximal, characteristic
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1987C211yesmaximal, characteristic, complemented
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1894C211yesmaximal, characteristic
15C2×#1894C211yesmaximal, characteristic
16C2×#1851C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2492—24no—
34#2492C2218yes—
35#2489—24no—
36#2489C2218yes—
37C2×#400—21no—
38C2×#400—21no—
39C2×#397—21no—
40C2×#393—21no—
41C2×#391—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48C2×#382—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2077C2218yes—
55#2056C2218yescomplemented
56#2056C2218yescomplemented
57#2026C2218yescomplemented
58#2024C2218yescomplemented
59#2001C2218yes—
60#1989C2218yescomplemented
61#1987C2218yescomplemented
62#1927C2218yescomplemented
63#1927C2218yescomplemented
64#1894C2218yes—
65#1894C2218yes—
66#1851C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×Q16—42no—
82C22×Q16—42no—
83C22×Q16—42no—
84C22×Q16C2314yes—
85C22×M16—42no—
86C22×M16—42no—
87C22×M16—42no—
88C22×M16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8C2314yes—
97C2×#125—24no—
98C2×#125—22no—
99C2×#125—42no—
100C2×#125C2314yes—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—42no—
106C2×C22⋊Q8—42no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C2×#106—42no—
131C2×#106C2314yes—
132C2×C4⋊C8—42no—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8—42no—
149C2×C22⋊C8C2314yes—
150#400—28no—
151#400—28no—
152#397—28no—
153#393—28no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—28no—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#391C2314yes—
163#388—28no—
164#388—28no—
165#382—28no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4D414yes—
175C4×C22⋊C4C2314yes—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—82no—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C2×Q16—44no—
244C2×Q16—44no—
245C2×Q16—44no—
246C2×Q16—416no—
247C2×M16—44no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—28no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270#125—28no—
271#125—44no—
272#125—48no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—82no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#106—44no—
338C4⋊C8—44no—
339C4⋊C8—44no—
340#103—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346C22⋊C8—44no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95C2×D414yes—
360#95C2×D414yes—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—84no—
399C2×Q8—816no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4#13714yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473Q16—416no—
474M16—416no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—44no—
489C2×C8—44no—
490C2×C8—44no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4C2×#4114yes—
513C4⋊C4C2×#4114yes—
514C22⋊C4—28no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—48no—
527C42—84no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23#53712yes—
543C23#54612yes—
544C23#54612yes—
545C23C2×#13712yes—
546C23#57512yes—
547C23#57712yes—
548C23#57812yes—
549C23C23×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—816no—
559Q8—84no—
560Q8—84no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4#53712yes—
618C2×C4#54214yes—
619C2×C4#54214yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#55914yes—
623C2×C4H(3,𝔽4)14yes—
624C2×C4C2×#13714yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C8—42no—
632C8—42no—
633C8—42no—
634C8—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—88no—
643C22#271214yes—
644C22#325214yes—
645C22#325814yes—
646C22#331214yes—
647C22#331314yes—
648C22#332314yes—
649C22#333114yes—
650C22C2×#53711yescharacteristic
651C22C2×#54611yescharacteristic
652C22C2×#54611yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57711yescharacteristic
656C22C2×#57811yescharacteristic
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#271214yes—
671C4C2×#54211yescharacteristic
672C4C2×#54211yescharacteristic
673C4C2×#55911yescharacteristic
674C4C2×H(3,𝔽4)11yescharacteristic
675C2—82no—
676C2#3212318yesdirect product factor
677C2C2×#271211yescharacteristic
678C2C2×#325211yescharacteristic
679C2C2×#325811yescharacteristic
680C2C2×#331211yescharacteristic
681C2C2×#331311yescharacteristic
682C2C2×#332311yescharacteristic
683C2C2×#333111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #32123C2direct (N × H)
5. C2×#2056C2semidirect
6. C2×#2056C2semidirect
7. C2×#2026C2semidirect
8. C2×#2024C2semidirect
10. C2×#1989C2semidirect
11. C2×#1987C2semidirect
12. C2×#1927C2semidirect
13. C2×#1927C2semidirect
55. #2056C22semidirect
56. #2056C22semidirect
57. #2026C22semidirect
58. #2024C22semidirect
60. #1989C22semidirect
61. #1987C22semidirect
62. #1927C22semidirect
63. #1927C22semidirect
676. C2#32123direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24892
C2×#24922
#321232
C2×#20772
C2×#20562
C2×#20262
C2×#20242
C2×#20012
C2×#19892
C2×#19872
C2×#19272
C2×#18942
C2×#18512